Function Repository Resource:

LocalExtrema

Source Notebook

Compute the local extrema of an expression with respect to the given variables

Contributed by: Wolfram|Alpha Math Team

ResourceFunction["LocalExtrema"][expr,x]

computes the local maxima and minima of expr with respect to x.

ResourceFunction["LocalExtrema"][expr,{x,y,…}]

computes the local maxima and minima of expr with respect to multiple variables.

ResourceFunction["LocalExtrema"][{expr,const},{x,y,…}]

computes the local maxima and minima of expr subject to the constraint const.

Details and Options

ResourceFunction["LocalExtrema"] returns an association of the form <|"Maxima"→{f_max,{x→x_max,y→y_max,…}},"Minima"→{f_min,{x→x_min,y→y_min,…}}|>.

The const can contain equations, inequalities or logical combinations of these.

ResourceFunction["LocalExtrema"] only returns results when there is a bounded extremum.

Examples

Basic Examples (2)

Compute the local extrema of a curve:

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Plot them:

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Use a constraint in order to reduce the domain upon which extrema can be found:

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Scope (2)

Compute the extrema of a function of two variables:

In[4]:=

$ResourceFunction["LocalExtrema"][(4 - x^2 - y^2)^2, {x, y}]$

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Compute the extrema of a piecewise function:

In[5]:=

$ResourceFunction["LocalExtrema"][ Piecewise[{{x, x <= 0}, {(x - 1)^2 - 4, x > 0}}], x]$

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Plot the result:

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$Plot[Piecewise[{{x, x <= 0}, {(x - 1)^2 - 4, x > 0}}], {x, -1, 2}, Epilog -> {Directive[PointSize[Large], Red], Point[{1, -4}], Point[{0, 0}]}]$

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Possible Issues (1)

LocalExtrema may return duplicate results for periodic functions:

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Publisher

Wolfram|Alpha Math Team

Version History

2.0.0 – 23 March 2023
1.0.0 – 22 September 2020

Related Resources

Author Notes

To view the full source code for LocalExtrema, run the following code: